Basis Vector Matrix and Vector Space / Subspace Spanned by a Basis Vector Matrix
Any \(M \times N\) Matrix (where \(M \geq 2\) and \(N \geq 2\) and \(M \geq N\)) containing only Linearly Independent Vectors as Columns is called a Basis Vector Matrix and the Vectors/Columns themselves are called Basis Vectors.
An \(M \times N\) Basis Vector Matrix consists of \(N\) Basis Vectors with each Basis Vector having \(M\) Components.
The value of \(M\) (i.e. Number of Components in each Basis Vector) gives the Dimension of the Basis Vector Matrix.
The value of \(N\) (i.e. Number Basis Vectors) gives the Span of the Basis Vector Matrix.
The Span of any Basis Vector Matrix is always Lesser or Equal to the Dimension of the Basis Vector Matrix (i.e. \(N \leq M\)).
If \(M = N\) (i.e. if the Span of the Basis Vector Matrix is Same as the Dimension of the Basis Vector Matrix), then the Basis Vector Matrix (i.e. the Set of Basis Vectors in the Basis Vector Matrix) is said to be Spanning a Complete/Full Vector Space of \(N\) Dimensions (or \(M\) Dimensions).
If \(N < M\) (i.e. the Span of the Basis Vector Matrix is Lesser Than the Dimension of the Basis Vector Matrix), then the Basis Vector Matrix (i.e. the Set of Basis Vectors in the Basis Vector Matrix) is said to be Spanning an \(N\) Dimensional Vector Subspace of an \(M\) Dimensional Vector Space.
Any 2 \(M \times N\) Basis Vector Matrices \(A\) and \(B\) Represent/Span the same Vector Space or Vector Subspace if all the Basis Vectors/Columns of Basis Vector Matrix \(A\) belong to the Column Space of the Basis Vector Matrix \(B\) and vice versa.